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Theorem imasf1oxmet 24336

Description: The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
imasf1oxmet.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasf1oxmet.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasf1oxmet.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasf1oxmet.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasf1oxmet.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
imasf1oxmet.d	⊢ 𝐷 = (dist‘𝑈)
imasf1oxmet.m	⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))

**Assertion**
Ref	Expression
imasf1oxmet	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Proof of Theorem imasf1oxmet Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1oxmet.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasf1oxmet.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasf1oxmet.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		f1ofo 6791	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
6		imasf1oxmet.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7		eqid 2737	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
8		imasf1oxmet.d	. . . 4 ⊢ 𝐷 = (dist‘𝑈)
9	1, 2, 5, 6, 7, 8	imasdsfn 17449	. . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
10	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “_s 𝑅))
11	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
12	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵)
13	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍)
14		imasf1oxmet.e	. . . . . . . 8 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
15		imasf1oxmet.m	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉))
17		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
18		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
19	10, 11, 12, 13, 14, 8, 16, 17, 18	imasdsf1o 24335	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
20		xmetcl 24292	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ^*)
21	20	3expb 1121	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
22	15, 21	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
23	19, 22	eqeltrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
24	23	ralrimivva 3181	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
25		f1ofn 6785	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
26	3, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
27		oveq2 7378	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏)))
28	27	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
29	28	ralrn 7044	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
31		forn 6759	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
32	5, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
33	32	raleqdv 3298	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
34	30, 33	bitr3d 281	. . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
35	34	ralbidv 3161	. . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
36	24, 35	mpbid 232	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*)
37		oveq1 7377	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦))
38	37	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
39	38	ralbidv 3161	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
40	39	ralrn 7044	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
41	26, 40	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
42	32	raleqdv 3298	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
43	41, 42	bitr3d 281	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
44	36, 43	mpbid 232	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*)
45		ffnov 7496	. . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
46	9, 44, 45	sylanbrc 584	. 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ^*)
47		xmeteq0 24299	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
48	16, 17, 18, 47	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
49	19	eqeq1d 2739	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝑎𝐸𝑏) = 0))
50		f1of1 6783	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–1-1→𝐵)
51	3, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝐵)
52		f1fveq 7220	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
53	51, 52	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
54	48, 49, 53	3bitr4d 311	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
55	16	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐸 ∈ (∞Met‘𝑉))
56		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
57	17	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑎 ∈ 𝑉)
58	18	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑏 ∈ 𝑉)
59		xmettri2 24301	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
60	55, 56, 57, 58, 59	syl13anc 1375	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
61	19	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
62	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑈 = (𝐹 “_s 𝑅))
63	11	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
64	12	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐹:𝑉–1-1-onto→𝐵)
65	13	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑅 ∈ 𝑍)
66	62, 63, 64, 65, 14, 8, 55, 56, 57	imasdsf1o 24335	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑎)) = (𝑐𝐸𝑎))
67	62, 63, 64, 65, 14, 8, 55, 56, 58	imasdsf1o 24335	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑏)) = (𝑐𝐸𝑏))
68	66, 67	oveq12d 7388	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) = ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
69	60, 61, 68	3brtr4d 5132	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
70	69	ralrimiva 3130	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
71		oveq1 7377	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑎)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑎)))
72		oveq1 7377	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑏)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))
73	71, 72	oveq12d 7388	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) = (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
74	73	breq2d 5112	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
75	74	ralrn 7044	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
76	26, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
77	32	raleqdv 3298	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
78	76, 77	bitr3d 281	. . . . . . . 8 ⊢ (𝜑 → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
79	78	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
80	70, 79	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
81	54, 80	jca 511	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
82	81	ralrimivva 3181	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
83	27	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0))
84		eqeq2 2749	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
85	83, 84	bibi12d 345	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ↔ (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏))))
86		oveq2 7378	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → (𝑧𝐷𝑦) = (𝑧𝐷(𝐹‘𝑏)))
87	86	oveq2d 7386	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
88	27, 87	breq12d 5113	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
89	88	ralbidv 3161	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
90	85, 89	anbi12d 633	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑏) → (((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
91	90	ralrn 7044	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
92	26, 91	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
93	32	raleqdv 3298	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
94	92, 93	bitr3d 281	. . . . 5 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
95	94	ralbidv 3161	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
96	82, 95	mpbid 232	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
97	37	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷𝑦) = 0))
98		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥 = 𝑦 ↔ (𝐹‘𝑎) = 𝑦))
99	97, 98	bibi12d 345	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦)))
100		oveq2 7378	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑧𝐷𝑥) = (𝑧𝐷(𝐹‘𝑎)))
101	100	oveq1d 7385	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))
102	37, 101	breq12d 5113	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
103	102	ralbidv 3161	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
104	99, 103	anbi12d 633	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
105	104	ralbidv 3161	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
106	105	ralrn 7044	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
107	26, 106	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
108	32	raleqdv 3298	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
109	107, 108	bitr3d 281	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
110	96, 109	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
111	15	elfvexd 6880	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
112		focdmex 7912	. . . 4 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
113	111, 5, 112	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
114		isxmet 24285	. . 3 ⊢ (𝐵 ∈ V → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
115	113, 114	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
116	46, 110, 115	mpbir2and 714	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 class class class wbr 5100 × cxp 5632 ran crn 5635 ↾ cres 5636 Fn wfn 6497 ⟶wf 6498 –1-1→wf1 6499 –onto→wfo 6500 –1-1-onto→wf1o 6501 ‘cfv 6502 (class class class)co 7370 0cc0 11040 ℝ^*cxr 11179 ≤ cle 11181 +_𝑒 cxad 13038 Basecbs 17150 distcds 17200 “_s cimas 17439 ∞Metcxmet 21311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-sup 9359 df-inf 9360 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-fz 13438 df-fzo 13585 df-seq 13939 df-hash 14268 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-0g 17375 df-gsum 17376 df-xrs 17437 df-imas 17443 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-mulg 19015 df-cntz 19263 df-cmn 19728 df-xmet 21319

This theorem is referenced by: imasf1omet 24337 xpsxmet 24341 imasf1obl 24449 imasf1oxms 24450

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